Forces between multiple charges and continuous charge distribution
The net force on a charge is the vector sum of the forces from each other charge acting alone. Spread-out charge is described by charge density.
The superposition principle
When many charges act on one charge, the force between any two of them is not changed by the others. So we find each force separately with Coulomb's law. The net force is the vector sum of these forces. This is the superposition principle. Forces must be added as vectors, by the parallelogram law or by components. For two forces F₁ and F₂ with an angle θ between them, the resultant is F = √(F₁² + F₂² + 2F₁F₂ cos θ).
Method for any net-force problem
- 1Draw the figure. Mark every charge, its sign and the distances.
- 2Find the size of each force on the chosen charge using F = kq₁q₂/r².
- 3Draw each force: away from a like charge, towards an unlike charge.
- 4Add the forces as vectors: use the parallelogram law, or resolve them into x and y components.
- 5Give the size of the resultant and its direction, as an angle or in words.
Worked example: equilateral triangle
Three charges of +2 μC each sit at the corners of an equilateral triangle of side 10 cm in air. Find the net force on any one charge.
- Each pair is 0.10 m apart. Force from one other charge: F = 9 × 10⁹ × (2 × 10⁻⁶)² ÷ (0.10)².
- F = 9 × 10⁹ × 4 × 10⁻¹² ÷ 0.01 = 0.036 ÷ 0.01 = 3.6 N.
- The two forces on the chosen charge are both repulsive, with 60° between them.
- Resultant = √(F² + F² + 2F² cos 60°) = √(3F²) = √3 F.
- Resultant = 1.732 × 3.6 N ≈ 6.2 N.
About 6.2 N on each charge, along the bisector of the angle at that corner, pointing away from the opposite side
Worked example: where is the net force zero?
A charge of +4 μC is at A and a charge of +1 μC is at B, 30 cm away. Where should a third charge be placed so that the net force on it is zero?
- The two charges are alike, so the point lies between A and B, nearer the smaller charge.
- Let the point be x from A. The two forces must be equal: k(4 μC)q/x² = k(1 μC)q/(0.30 − x)².
- So 4/x² = 1/(0.30 − x)². Take square roots: 2/x = 1/(0.30 − x).
- 2(0.30 − x) = x, so 0.60 = 3x and x = 0.20 m.
20 cm from A (10 cm from B), on the line between them; the answer does not depend on the size or sign of the third charge
Continuous charge distributions
| Type | Charge density | SI unit | Example | Charge on a small element |
|---|---|---|---|---|
| Along a line | λ (lambda), charge per unit length | C m⁻¹ | A thin charged wire or ring | dq = λ dl |
| Over a surface | σ (sigma), charge per unit area | C m⁻² | A charged metal plate, or the surface of a charged sphere | dq = σ dS |
| Through a volume | ρ (rho), charge per unit volume | C m⁻³ | A charged non-conducting solid ball | dq = ρ dV |
Why we use charge densities
A charged body holds billions of electrons packed close together. Treating each one separately is impossible. So we imagine the charge spread smoothly and describe it by a density. We cut the body into tiny elements. Each element carries a small charge dq and acts like a point charge. The total force or field is found by adding (integrating) the effects of all the elements. This is superposition again.
Computer programs that design new medicines add up Coulomb forces from thousands of charged atoms to see how a drug fits a protein. They use exactly this superposition principle, one pair of charges at a time.
3 marks: “Charges +q, +q and −q sit at the corners of an equilateral triangle of side a. Find the net force on the charge −q.” Each +q attracts −q with F = kq²/a², and the two forces are 60° apart. Resultant = √3 F = √3 kq²/a², directed from −q towards the midpoint of the line joining the two +q charges.
1 marks: “Define surface charge density and give its SI unit.” Charge per unit area, σ = ΔQ/ΔS; unit C m⁻².
2 marks: “Charges +9 μC and +1 μC are 40 cm apart. Where will a third charge feel no net force?” Between them, x from the 9 μC charge: 3/x = 1/(0.40 − x), so x = 0.30 m. The point is 30 cm from the 9 μC charge (10 cm from the 1 μC charge).
Forces from many charges: one at a time, then add as arrows (vectors), never as plain numbers.